In consoles, why are BF4 and BL2 aiming much more free than other games?

So I have played a few FPS games on console, including Bioshock, Crysis, CoD, and Destiny. However, the two games that I noticed a pretty big difference in aiming are Borderlands 2, Minecraft, and Battlefield 4.

In the games I listed before, it seems like the aim starts off slowly and accelerates the more you hold it in one direction. Also, I had to move the DS3 R3 analog further distances in order for the aim to start moving. The same went for DS4 when I played Destiny and CoD. This feels a bit noobish, and I felt like I had no experience in FPS whatsoever (especially being a former PC gamer)

In BF4, MC and Borderlands 2, however, I noticed that the least that I moved the stick, the aim already started moving. I noticed it more on BF4 in the PS4, because I played BL2 with the DS3, an inferior controller to the DS4 in my opinion.

Also in these games, the aim does not seem to "accelerate" like in the other ones, and I felt like I had to accelerate it myself. That made me feel more like a pro.

Has anyone else noticed this in consoles? Is there a way to bypass this aim type and make games feel more like BF4? Why isn't this an option rather than the default in games?

I have seen a similar option with the mouse every time I tweaked the settings on PC, but only in a few games. Like in PC Minecraft there's a similar setting called "cinematic camera."

Because not all shooters are designed to play the same way. Some games intentionally give your weapons a "weightier" feel than others. It's a design decision so that not every gun in every game feels the same.

It's pretty much all a matter of personal preference.